## 59.11 Sheaves

Definition 59.11.1. A presheaf $\mathcal{F}$ of sets (resp. abelian presheaf) on a site $\mathcal{C}$ is said to be a separated presheaf if for all coverings $\{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov} (\mathcal{C})$ the map

$\mathcal{F}(U) \longrightarrow \prod \nolimits _{i\in I} \mathcal{F}(U_ i)$

is injective. Here the map is $s \mapsto (s|_{U_ i})_{i\in I}$. The presheaf $\mathcal{F}$ is a sheaf if for all coverings $\{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov} (\mathcal{C})$, the diagram

59.11.1.1
$$\label{etale-cohomology-equation-sheaf-axiom} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j \in I} \mathcal{F}(U_ i \times _ U U_ j), }$$

where the first map is $s \mapsto (s|_{U_ i})_{i\in I}$ and the two maps on the right are $(s_ i)_{i\in I} \mapsto (s_ i |_{U_ i \times _ U U_ j})$ and $(s_ i)_{i\in I} \mapsto (s_ j |_{U_ i \times _ U U_ j})$, is an equalizer diagram in the category of sets (resp. abelian groups).

Remark 59.11.2. For the empty covering (where $I = \emptyset$), this implies that $\mathcal{F}(\emptyset )$ is an empty product, which is a final object in the corresponding category (a singleton, for both $\textit{Sets}$ and $\textit{Ab}$).

Example 59.11.3. Working this out for the site $X_{Zar}$ associated to a topological space, see Example 59.10.3, gives the usual notion of sheaves.

Definition 59.11.4. We denote $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ (resp. $\textit{Ab}(\mathcal{C})$) the full subcategory of $\textit{PSh}(\mathcal{C})$ (resp. $\textit{PAb}(\mathcal{C})$) whose objects are sheaves. This is the category of sheaves of sets (resp. abelian sheaves) on $\mathcal{C}$.

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